Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196266 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35250 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,13,16,11)(8,17)(9,18)(10,12)(14,15,20,19);(1,2)(3,4)(5,7)(6,11)(8,14)(9,19)(10,16)(12,13)(15,17)(18,20);(2,4)(5,8,17,6,10,18,9,12)(7,14,15,11,16,20,19,13);(5,7)(6,11)(8,14)(9,19)(10,16)(12,13)(15,17)(18,20)Polycyclic presentation:
2^9|a^2=h,c^2=fhi,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,f^b=fi,h^b=hi,e^c=ei,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72252449, D4×#1910, #72252459, … |
| maximal normal subgroup of | #72252449, D4×#1910, #72252459, … |
| maximal quotient of | #72260439, #72260853 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281240);